The classical Artin-Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations.We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.6 See [Pre85, p. 354], also reproven in [FHV94, Corollary 1.3]. 7 The PRC property can indeed be seen as a very strong independence assumption, since it implies in particular that distinct orderings on K induce distinct topologies, cf. [Pre85, p. 353].8 See discussion in Section 5.7 how this follows from the results in [Ers01]. 9 We use "compact" to mean what other sources call "quasi-compact", i.e. there is no implication of being a Hausdorff space.